A gravity gradiometer is an instrument which is capable of measuring one or more components of the spatial gradient of the gravitational specific force vector.
As Sir Isaac Newton first showed in his Philosophiæ Naturalis Principia Mathematica (The Royal Society, London 1867), at every point in space there exists a vector field (the gravitational field) which exerts a force on any body located at that point which is proportional to the mass of that body. That force is a vector quantity, with three orthogonal components. Denoting the value of that vector at a location in space as , and denoting a reference frame A as A (using the vectrix notation from Appendix B of [Hughes, P. C., Spacecraft Attitude Dynamics, John Wiley & Sons, Toronto, 1986]), that vector can be expressed as a 3-element column vector by projecting it onto reference frame A, thusly:
                              a          A                =                                            ℱ                              A                ⁢                                                                                        ·                          a              _                                ⁢                      =            Δ                    ⁢                      [                                                                                a                                          A                      x                                                                                                                                        a                                          A                      y                                                                                                                                        a                                          A                      z                                                                                            ]                                              (        1        )            where, as shown in FIG. 1b, A has as its three orthogonal reference axes the unit vectors ,  and  (referred to here as the x, y and z axes of reference frame A), and aAx, aAy and aAz are the projections of the vector  onto those unit vectors.
The gravity gradient, which is the spatial gradient of the vector , is denoted here as , and is a second-order tensor. This tensor can be projected onto reference frame A to form a 3×3 matrix ΓA, thusly:
                              Γ          A                =                                            ℱ              A                        ·                          Γ              _                        ·                          ℱ              A              T                                ⁢                      =            Δ                    ⁢                      [                                                                                Γ                                          A                      xx                                                                                                            Γ                                          A                      xy                                                                                                            Γ                                          A                      xz                                                                                                                                        Γ                                          A                      yx                                                                                                            Γ                                          A                      yy                                                                                                            Γ                                          A                      yz                                                                                                                                        Γ                                          A                                              zx                        ⁢                                                                                                                                                                                                            Γ                                          A                      zy                                                                                                            Γ                                          A                      zz                                                                                            ]                                              (        2        )            where the elements of ΓA are the partial derivatives of aAx, aAy and aAz with respect to x, y and z. That is, ΓAxx=∂aAx/∂x, ΓAxy=∂aAx/∂y, ΓAxz=∂aAx/∂z, ΓAyx=∂aAy/∂x, ΓF*Ayy=∂aAy/∂y, ΓAyz=∂aAy/∂z, ΓAzx=∂aAz/∂x, ΓAzy=∂aAz/∂y and ΓAzz=∂aAz/∂z. Note that as a conservative field, two fundamental properties of that field are that the matrix ΓA is symmetric, that is, ΓAxy=ΓAyx, ΓAxz=ΓAzx and ΓAyz=ΓAzy, and that it satisfies Laplace's equation, with the result that the trace of ΓA is equal to zero, i.e., ΓAxx+ΓAyy+ΓAzz=0.
A gravity gradiometer is thus an instrument which, when oriented in a particular way relative to (for example) reference frame A, measures one of the components of the matrix ΓA, or some combination of the components of ΓA. Gravity gradiometers of several different types have been developed, including the torsion balance of Eötvös (the first gravity gradiometer, which measures a combination of ΓAxy, ΓAxx−ΓAyy, ΓAxz and ΓAyz), the gravity gradiometer of Lancaster-Jones (which measures a combination of ΓAxz and ΓAyz), the rotating gravity gradiometer of Forward (which measures a combination of ΓAxy and ΓAxx−ΓAyy, as described in de Bra, D. B., Harrison, J. C. & Muller, P. M., “A proposed Lunar orbiting gravity gradiometer experiment,” The Moon, Volume 4, Issue 1-2, pp. 103-112), the rotating gravity gradiometer of Metzger (which measures a combination of ΓAxy and ΓAxx−ΓAyy, as described in Metzger, E. H., “Development experience of gravity gradiometer system,” IEEE Position Location and Navigation Symposium 1982, PLANS 198, pp. 323-332), the cryogenic orthogonal quadrupole responder gravity gradiometer of van Kann (which measures a combination of ΓAxy and ΓAxxΓAyy, as described in Matthews, R., “Mobile Gravity Gradiometry,” Ph. D. thesis, University of Western Australia, 2002), the cryogenic orthogonal quadrupole responder gravity gradiometer of Paik and Moody (which measures a combination of ΓAxy and ΓAxx−ΓAyy), the GOCE satellite's electrostatically levitated gravity gradiometer of Touboul et al. (which measures combinations of all of the components of ΓA, as described in Drinkwater, M. R., Flòberhagen, R. F., Haagmans, R., Muzi, D. & Popescu, A., “GOCE: ESA's First Earth Explorer Core Mission,” Space Sci. Rev., Vol. 108, 2003, pp. 419-432), and the atom-interferometer gravity gradiometer of Kasevich (which measures ΓAzz, as described in McGuirk, J. M., Foster, G. T., Fixler, J. B., Snadden, M. J. & Kasevich, M. A., “Sensitive absolute-gravity gradiometry using atom interferometry,” Phys. Rev. A, Vol. 65, 033608 (2002)).